QUANTUM STRINGS AND RANDOM SURFACES
217
The total number of these states is given by the number of ways in
which the integer n can be split into a sum of positive integers. Indeed, if
it were not for the space-time indices /x, we would have the general state
in the form:
|/> = a,-;
p>
/ j -f ... + = X I, Ij > 0
The number of such states is given by:
N(n) =;
1
dx
.n + 1 n 1
fc = l ^
1
27ri T
1 — x''
To derive this, let us represent (9.293) as
(9.293)
(9.294)
Then
L kPk = n
N(n) = I
{Pk}
2m J x
I
^ 1 r dx
1
2ot'-^x"+‘ M 1 - x ‘
When we add ^ space-time dimensions it is clear, that the number of
states N^(n) at the n-th level is given by:
I N^(n)x" = n
^
n=0
\k=l ^
^
1
(9.295)
Now, not all states (9.292) are physical. We have to apply the condition
T„\f} = 0. This results in equations for
etc. similar to (9.287).
The number of such states will be generated by:
n
1
= X N f’ '\n)x"
M = 0
1 - x\
Nfy\n) = N^_,(n)
(9.296)